7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves an exponential function, e^(3 - 4x). To simplify, identify the inner function (3 - 4x) and its derivative (-4). This suggests using substitution.

Step 2: Perform substitution. Let u = 3 - 4x, which implies that du/dx = -4 or equivalently, dx = -du/4.

Step 3: Rewrite the integral in terms of u. Substituting u and dx, the integral becomes ∫ e^u * (-du/4). Factor out the constant -1/4 to simplify: (-1/4) ∫ e^u du.

Step 4: Integrate e^u with respect to u. The integral of e^u is simply e^u, so the result becomes (-1/4) * e^u + C, where C is the constant of integration.

Step 5: Substitute back u = 3 - 4x to return to the original variable. The final expression is (-1/4) * e^(3 - 4x) + C.

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